Comment on “On the polynomial integrability of the Kirchoff equations, Physica D 241 (2012) 1417–1420”

Llibre and Valls, in [Physica D, 241(2012) 1417–1420], proved that, if the Kirchoff equations have a proper Darboux polynomial with its cofactor satisfying some symmetry, they have a polynomial first integral. In this note we will improve this last result, and obtain that, if the Kirchoff equations have a proper Darboux polynomial, they always have a polynomial first integral functionally independent of the three known ones. Our result improves that of Llibre and Valls in two aspects: we drop the symmetric condition, and prove that the obtained first integral is functionally independent of the known ones.

► We prove the existence of the fourth polynomial first integral of Kirchoff’s equations.
► The result improves the corresponding known one in two aspects, as follows.
► We drop the symmetric condition.
► We prove the functional independence of the first integral with the known three ones.